THE ALTERNATIVE DUNFORD-PETTIS PROPERTY IN SUBSPACES OF OPERATOR IDEALS

Title & Authors
THE ALTERNATIVE DUNFORD-PETTIS PROPERTY IN SUBSPACES OF OPERATOR IDEALS

Abstract
For several Banach spaces X and Y and operator ideal $\small{\cal{U}}$, if $\small{\cal{U}}$(X, Y) denotes the component of operator ideal $\small{\cal{U}}$; according to Freedman`s definitions, it is shown that a necessary and sufficient condition for a closed subspace $\small{\cal{M}}$ of $\small{\cal{U}}$(X, Y) to have the alternative Dunford-Pettis property is that all evaluation operators $\small{\phi_x\;:\;\cal{M}\;{\rightarrow}\;Y}$ and $\small{\psi_{y^*}\;:\;\cal{M}\;{\rightarrow}\;X^*}$ are DP1 operators, where $\phi_x(T)\; Keywords Dunford-Pettis property;Schauder decomposition;compact operator;operator ideal; Language English Cited by 1. Strongly alternative Dunford–Pettis subspaces of operator ideals, Ukrainian Mathematical Journal, 2013, 65, 4, 649 References 1. M. D. Acosta and A. M. Peralta, The alternative Dunford-Pettis property for subspaces of the compact operators, Positivity 10 (2006), no. 1, 51–63. 2. J. Becerra Guerrero and A. M. Peralta, The Dunford-Pettis and the Kadec-Klee properties on tensor products of$JB^{\ast}-triples$, Math. Z. 251 (2005), no. 1, 117–130. 3. S. W. Brown, Weak sequential convergence in the dual of an algebra of compact operators, J. Operator Theory 33 (1995), no. 1, 33–42. 4. L. J. Bunce and A. M. Peralta, Dunford-Pettis properties, Hilbert spaces and projective tensor products, J. Funct. Anal. 253 (2007), no. 2, 692–710. 5. L. J. Bunce and A. M. Peralta, The alternative Dunford-Pettis property, conjugations and real forms of$C^{\ast}-algebras$, J. London Math. Soc. (2) 71 (2005), no. 1, 161–171. 6. L. J. Bunce and A. M. Peralta, The alternative Dunford-Pettis property in$C^{\ast}-algebras$and von Neumann preduals, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1251–1255. 7. J. M. F. Castillo and M. Gonzalez, New results on the Dunford-Pettis property, Bull. London Math. Soc. 27 (1995), no. 6, 599–605. 8. C. H. Chu and B. Iochum, The Dunford-Pettis property in$C^{\ast}-algebras\$, Studia Math. 97 (1990), no. 1, 59–64.

9.
A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Mathematics Studies, 176. North-Holland Publishing Co., Amsterdam, 1993.

10.
J. Diestel, A survey of results related to the Dunford-Pettis property, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979), pp. 15–60, Contemp. Math., 2, Amer. Math. Soc., Providence, R.I., 1980.

11.
J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, 92. Springer-Verlag, New York, 1984.

12.
W. Freedman, An alternative Dunford-Pettis property, Studia Math. 125 (1997), no. 2, 143–159.

13.
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I, II, Springer- Verlag, Berlin, 1996.

14.
S. M. Moshtaghioun and J. Zafarani, Weak sequential convergence in the dual of operator ideals, J. Operator Theory 49 (2003), no. 1, 143–151.

15.
A. M. Peralta and I. Villanueve, The alternative Dunford-Pettis property on projective tensor products, Math. Z. 252 (2006), no. 4, 883–897.

16.
A. Pietsch, Operator Ideals, North-Holland Mathematical Library, 20. North-Holland Publishing Co., Amsterdam-New York, 1980.

17.
E. Saksman and H. O. Tylli, Structure of subspaces of the compact operators having the Dunford-Pettis property, Math. Z. 232 (1999), no. 3, 411–425.

18.
A. Ulger, Subspaces and subalgebras of K(H) whose duals have the Schur property, J. Operator Theory 37 (1997), no. 2, 371–378.